| Thu, Feb 05, 2026
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Computational and Applied Math Seminar
3:00 PM
MSCS 514 | | Rigidity for homogeneous solutions to the two-dimensional stationary Euler equations
Xukai Yan, Oklahoma State University
| | | Abstract: I this talk, I will talk about the rigidity problem of special solutions for the 2D incompressible stationary Euler equations. The main question is: if a solution is in a specific form on the boundary of a domain, must it take this form inside the domain? Hamel and Nadirashvili established the rigidity of shear flows in $2$D strips, half-plane and the whole plane, as well as rigidity of circular flows in annuli, exterior of disks, punctured disks, etc. The rigidity problem has also been studied for other fluid equations such as Boussinesq equations, MHD equations and Navier-Stokes equations. In this talk, I will discuss rigidity of homogeneous solutions of $2$D stationary Euler equations. I will first and talk about the background of the problem and some results on rigidity of shear flows and circular flows. I will also describe known results on the existence and classification of homogeneous solutions of stationary Euler equations. I will then present a recent work on rigidity of homogeneous solutions in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where $0\le a < b \le +\infty$ and $0< \theta_0 \le 2\pi$. In particular, we show that if a solution satisfies some homogeneity assumptions on the boundary of the domain and its radial or angular component has no stagnation point inside the domain, then it must be homogeneous throughout the domain. This is a joint work with Li Li and Zhibo Yang. |
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